The following functional dependencies hold true for the relational schema R{V, W, X, Y, Z}:
V -> W
VW -> X
Y -> VX
Y -> Z
Which of the following is irreducible equivalent for this set of functional dependencies?
(A) A
(B) B
(C) C
(D) D
Answer: (A)
Explanation: Given
V -> W
VW -> X
Y -> VX
Y -> Z
We need to find the minimal cover of these FDs
Option B. W->X can  not implied by the given FDs, so incorrect
Option C. Y->X can be implied from Y->V and V->X, hence redundant
Option D. W->X can  not implied by the given FDs, so incorrect
Option A. Minimal cover of dependencies that can be extracted out as
- V -> W
- V -> X
- Y -> V
- Y -> Z
Therefore, option A is most appropriate.
Alternate Solution
Irreducible equivalent functional dependency is minimal cover.
Given,
{V → W, VW → X, Y → VX, Y → Z}
W extraneous from VW, since we have V → W,
= {V → W, V → X, Y → VX, Y → Z}
X extraneous from VX, since we have V → X,
= {V → W, V → X, Y → V, Y → Z}
This explanation is contributed by Mithlesh Upadhyay
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